1 / 22

MA557/MA578/CS557 Lecture 19

MA557/MA578/CS557 Lecture 19. Spring 2003 Prof. Tim Warburton timwar@math.unm.edu. Advection-Diffusion Equation. DG Scheme for the Scalar ADE (Lax-Friedrichs flux – equivalent to upwind). DG Derivative Operator.

sailor
Télécharger la présentation

MA557/MA578/CS557 Lecture 19

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MA557/MA578/CS557Lecture 19 Spring 2003 Prof. Tim Warburton timwar@math.unm.edu

  2. Advection-Diffusion Equation

  3. DG Scheme for the Scalar ADE(Lax-Friedrichs flux – equivalent to upwind)

  4. DG Derivative Operator • We are going to introduce a DG derivative operator to simplify the scheme definition. • The linear operator Dtilde is such that the following holds for all intervals Ij • With the choice of penalty terms tauL and tauR to be determined.

  5. In Operator Notation Advectionterm(Lax-Friedrichs flux ~ upwind fluxfor scalar case) Diffusionterm

  6. General Dtilde Operator • We previously defined Dtilde for the special case where the test function is compactly supported on the j’th cell. • We generalize by:

  7. Skew-symmetry of Dtilde1,1 [integrate by parts]

  8. Skew-symmetry • Note:the negative sign when the operator moves. • Trivially:

  9. Recap • We already proved that the DG scheme, with Lax-Friedrichs fluxes, for advection (D=0) is stable in the sense that : • Adding the diffusion term (dropping boundary terms) gives: • In terms of norms:

  10. Stability!! • Assuming D>=0 (reasonable since particles do not jump randomly with negative rates)

  11. D operator • If we move to the Legendre basis we can write down a discrete description of Dtilde:

  12. Implementation of Dtilde function [dfdx] = LEGdgderiv(f, nodex, tauL, tauR, D, F, G, H, J) [p,N] = size(f); % p = maximum polynomial order p=p-1; % N = number of nodes N=N+1; % space for derivative dfdx = zeros(p+1,N-1); % cells 2 to N-2 ids = (2:N-2); dfdx(:,ids) = (D*f(:,ids) + tauL*F*f(:,ids) + tauL*G*f(:,ids-1) + tauR*H*f(:,ids) + tauR*J*f(:,ids+1)); % cell 1 (periodicity assumed) dfdx(:,1) = (D*f(:,1) + tauL*F*f(:,1) + tauL*G*f(:,N-1) + tauR*H*f(:,1) + tauR*J*f(:,2)); % cell N-1 (periodicity assumed) dfdx(:,N-1) = (D*f(:,N-1) + tauL*F*f(:,N-1) + tauL*G*f(:,N-2) + tauR*H*f(:,N-1) + tauR*J*f(:,1)); % apply chain rule for physical cell width dx = nodex(2:N)-nodex(1:N-1); coeff = ones(p+1,1)*(2./dx); dfdx = dfdx.*coeff;

  13. In Action (advection eqn) % TIME STEPPING tauL = (1/u)*(u+abs(u))/2; tauR = (1/u)*(u-abs(u))/2; for tstep = 1:Ntsteps sigma = rho; for rkstage = 3:-1:1 dsigmadx = LEGdgderiv(sigma, nodex, tauL, tauR, D, F, G, H, J); sigma = rho + (dt/rkstage)*(-u*dsigmadx); end rho = sigma; if ( ~mod(tstep, 100) ) q = V*rho; % plot(x(:),abs(q(:)-exp(-(x(:)-dt*tstep).^2))); plot(x,q); pause(0.1); tstep*dt end end

  14. Discrete Scheme • Step 1: q = x DG derivative of C • Step 2: time rate of change of C Advectionterm(Lax-Friedrichs flux ~ upwind fluxfor scalar case) Diffusionterm

  15. Dropping q • We can do away with q: • This should make the following observation clear. The dt dependence for stability is now: • The first term is due to the spectral radius of the advection operator. • The second term is due to the spectral radius of the diffusion operator.

  16. In Action (ADE) % TIME STEPPING for tstep = 1:Ntsteps sigma = rho; for rkstage = p:-1:1 % advection: upwind bias on flux terms dsigmadx = LEGdgderiv(sigma, nodex, tauL, tauR, D, F, G, H, J); % diffusion: centered fluxes dsdx = LEGdgderiv(sigma, nodex, 1, 1, D, F, G, H, J); d2sdx2 = LEGdgderiv(dsdx, nodex, 1, 1, D, F, G, H, J); sigma = rho + (dt/rkstage)*(-u*dsigmadx+Dcoeff*d2sdx2); end rho = sigma; if ( ~mod(tstep, 400) ) q = V*rho; plot(x,q); axis([xmin xmax 0 1]) pause(0.1); end end

  17. Downloads • You can download the Dtilde based scripts from: • http://www.math.unm.edu/~timwar/MA578S03/MatlabScripts/LEGadvectionLaxF.m • http://www.math.unm.edu/~timwar/MA578S03/MatlabScripts/LEGade.m • You will need the Dtilde routine (named LEGdgderiv.m): • http://www.math.unm.edu/~timwar/MA578S03/MatlabScripts/LEGdgderiv.m

  18. Legendre Scheme To Lagrange Scheme(modal to nodal) • So far we have used Legendre expansions to represent the solution in each cell. • This representation causes some problems if we wish to evaluate non-linear function of the solution. • For instance solving the inviscid Burger’s equation:requires the evaluation of u2

  19. Transforming the Modal Advection DG Scheme To a Nodal Advection DG Scheme • Recall that given a Legendre expansion we may evaluate it at a set of points by: • We substitute: into the scheme:

  20. Nodal Scheme • Multiplying both sides on the left with V • This clearly shows that the nodal scheme is a change of basis away from the modal scheme.

  21. Summary of Nodal Scheme Note: we do not need to transform the initial condition or solution for visualization…

  22. Next Lecture • The lecture on Friday 03/07 is cancelled in favour of the Analysis Conference • The lecture on Monday 03/10 will be the first time we generalize the DG to two spatial dimensions. • Do not miss the Monday lecture  • Run the LEGade.m script (you will need the LEGdgderiv.m script as well) • How to use nodal methods in general.

More Related